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Questions tagged [calculus-of-variations]

Optimization of functionals mostly defined on infinite-dimensional spaces.

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38 views

Computing Euler Lagrange Equation for a Certain Functional

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where ...
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19 views

Continuous adjoint of the one-dimensional Laplace equation

Say I have a problem given by the 1D Laplace equation, $$ R (T(\alpha), \alpha) = \frac{d^2 T(x)}{dx^2} - \alpha(x) T (x) = 0, $$ with $x \in [0,1]$, Dirichlet boundary conditions on $x=0$ and $x=1$, ...
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1answer
28 views

“Quick” proof of the fundamental lemma of calculus of variations

Here's the statement: Let $f \in C([a,b])$ and $H$ be the set $\{h\in C([a,b]):h(a)=h(b)=0\}$. If $\int_a^bf(x)h(x)\,\text{d}x=0$ for all $h\in H$, then $f(x)=0$ for all $x\in [a,b]$. I saw a lot ...
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13 views

Using Calculus of Variations to Find the Maximum Second Moment of Area for Constrained Area

I'm trying to maximize centroidal second moment of area while area is constrained. We know, $$ I_{xx} = \int\limits_{D} (y^{2} - y_{c}^{2}) dA $$ Where $$y_{c}$$ is $$ \frac{\int\limits_{D} y dA}{A}$$...
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30 views

Lagrangian density of second order hyperbolic equation

It's been some hours now that I am trying to find the Lagrangian density of the following hyperbolic PDE with variable coefficients and $c_{ij}=c_{ji}(x)$ $$ \partial^2_t u - c_{ij}\partial_i\...
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1answer
41 views

Calculus of variations with differential forms

I want to generalize calculus of variations with differential forms. Or better, I saw it somewhere some time ago, but now I cannot re-build it. Here is what I remember. Let be $(M, I, \Lambda)$ a ...
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18 views

Initial conditions on Lagrange Multipliers in variational problems

If one wants to extremize the integral $\int_{x_0}^{x_1} F(x, y_i, y_i')dx$ subject to constraints $\phi_j = g_j(x,y_i,y_i')=0$, using the calculus of variations, then one can generate the Euler-...
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18 views

Existence of focal points : Semi-Riemannian Geometry

Let (M,g) Riemannian or Lorentz with P spacelike submanifold (immersed) in M and $\sigma:[0,\infty)\rightarrow M$ (unit) cospacelike geodesic normal to P at $p=\sigma(r)$ under the hypotheses: (1)$H(\...
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How to constrain function to be capped by a maximal value in a variational problem?

In a variational problem, where one seeks function $y(x)$ that is an extreme of a functional $$ J[y]=\int_a^b L(y(x), y'(x), x) \, dx\;, $$ one can constrain $y(x)$ by condition $$ C[y]=\int_a^b G(y(...
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18 views

Arclength integral result does not correspond to intuition

Consider the problem of finding the first variation $\delta_{y_0}J$ for $J(y)=\int_0^1\sqrt{1+y'(x)^2}\,\text{d}x$, given $y_0(x)=ax+b$. I don't know if I'm proceeding correctly and the answer I'm ...
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2answers
39 views

Euler–Lagrange equation has no solutions

Show that the Euler–Lagrange equation for the functional: $$I(y) = \int_{0}^{1}y dx$$ subject to y(0) = y(1) = 0 has no solutions. Explain why no extremum for I exists. When forming the E-L ...
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28 views

what does “scalar mean curvature of $\partial E$ with orientation induced by the inner normal to E” mean?

I'm currently reading the paper https://arxiv.org/pdf/1007.3899.pdf and I have a question regarding the mean curverture stuff in it (unfortunately, I don't have any knowledge in differential geometry)-...
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2answers
46 views

Euler-Lagrange equations for dependent multiple functions

Find the extremals for the functional: $$ J(x) = \int_{0}^{1}\left[x\left(t\right)\dot{x}\left(t\right) + \ddot{x}^{2}\left(t\right)\right]\mathrm{d}t $$ where $x(0)=0$, $\dot{x}(0)=1$, $x(1)=2$, $...
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17 views

About the Frechet derivative of a functional

How can I compute the Frechet derivative of the functional $$ I(u)=\frac{1}{2}\int_{\Omega} \vert \nabla u\vert ^2\ dx \ + \ \int_{\Omega}\left[1-|u|^2\right]^2\ dx$$ for $u$ in a functional ...
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1answer
163 views

Existence and Uniqueness of Poisson Equation with Robin Boundary Condition using First Variation Methods

I'm currently stuck on the following exercise from Evans PDE Chapter 8 Exercise 11. Let $\beta: \mathbb{R} \rightarrow \mathbb{R}$ be smooth with \begin{equation} 0 < a \leq \beta'(z) \leq b, \...
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1answer
51 views

Proving a Certain Functional is Coercive

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where ...
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11 views

Converse of the Extremization Condition in Calculus of Variations

My understanding of the calculus of variations is as follows. Let $U=(a,b)\subset\mathbb{R}$ be an open interval and let $L:(C^2(U))^{2n} \times U\to C^2(U)$ be at least $C^2$-differentiable with ...
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12 views

What's the expression of $\frac{\delta F}{\delta P_i}$

$$F=\int|\nabla \vec P|^2dv$$ $$\vec P=(P_x,P_y)$$ what's the expression of $\frac{\delta F}{\delta P_i}$,($i=x,y)$
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1answer
23 views

How to prove that $y$ is a minimiser for a certain functional $J$?

The problem Consider the functional $$J(y)=\int_a^b (y')^2 \, \mathrm{d}x$$ where $y \in D = C^{1,\mathrm{pw}}[a,b] \cap \{ y(a)=A,y(b)=B \}$ (the usual domain in the calculus of variations). We are ...
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A continuity result for functions on a Sobolev space

Let $$W^{1,p}_T = \{u \in W^{1,p}([0,T];\mathbb{R}^N) \mid u(0) = u(T)\},$$ where $W^{1,p}([0,T],\mathbb{R}^n)$ is the usual Sobolev space of functions from $[0,T]$ to $\mathbb{R}^N$. Let $$F:[0,T] \...
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1answer
42 views

Uniqueness of minimizer of Lagrangian

I have recently started calculus of variations and face the following task: Consider the Lagrangian $$L(y',y):=|y'|$$ on the space of curves $$U = \{y\in C^2([0;T];\mathbb R): y(0)=a, y(T=1)=b \}$$ ...
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15 views

Attaining extrema when stationarity condition has no solution

I was wondering if someone could shed some light on the following for me: If a stationarity (maximizing or minimizing) condition has no solution inside a particular domain, then how do we reason that ...
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35 views

Momentum constraints for a Singular Lagrangian

Note I've explicitly indicated it at points in this question, but unless stated otherwise $i,j,k \in \{1, \ldots, n\}$, $a,b,c \in \{1, \ldots, R_W\}$, and $\alpha, \beta, \gamma \in \{R_W + 1, \...
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1answer
29 views

Strictly convexity of $\rho \mapsto \int_{\mathbb{R}^N} |\nabla \sqrt{\rho}|^2 \,dx$

I want to know Is the map $\rho \mapsto \int_{\mathbb{R}^N} |\nabla \sqrt{\rho}|^2 $ from $X \to \mathbb{R}$ strictly convex? Here, $X = \{\rho\in W^{1,1}_+(\mathbb{R}^n)\mid \int \rho=1,\,\...
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17 views

How to take the determinant of a rank(1,1) tensor?

I want to find the Jacobian matrix and its determinant of the generic infinitesimal transformation: $x'^\mu=x^\mu+\epsilon_\alpha\frac{\delta x^\mu}{\delta \epsilon_\alpha}$ where $\epsilon_\alpha$ ...
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46 views

Paths on a disconnected manifold

I would like to check my understanding of the following. Let M be a smooth embedded submanifold of $R^n$, and let $q_1,q_2$ be points in M. Define "a path on M between $q_1, q_2$" as a $C^2$ map $p:...
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1answer
48 views

weak convergence of weak derivative in Sobolev Spaces

Short question cornerning some lectures notes in my current calculus of variation class: Let $\Omega \subset \mathbb{R^n}$ be open and bounded. It is now stated that if $(\phi_j)_{j \in \mathbb{N}} \...
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43 views

In order to solve Euler Equation i get 1=0 what that's means

So I tried to understand what happened if the Lagrangian $L$ which minimize the action $S$ $$S=\int{L(x,u(x)) d x}$$ Doesn't depend on $ u'(x) $ and depend only on $u(x)$ For example let's looking ...
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Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that, evaluated at $x=e^t$, it minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\...
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1answer
30 views

Finding y(x) and extremals of functional with Euler equation

I want to use the Euler equation to find $y=y(x)$ to get the extremals for the following functional: $$\int_a^b \left[xy + 2\left(\frac{dy}{dx}\right)^2\right] \, \mathrm{d}x$$ I know that the Euler ...
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1answer
49 views

Proving Young's Inequality using the Legendre Transform

In this question: Legendre transform and Young's Inequality, given $ f \colon \mathbb{R}^n \to \mathbb{R} $, $ f(x) = \frac{1}{p}|x|^{p} $ for $ 1 < p < \infty $, the Legendre transform of $ f $ ...
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77 views

Doubt about second variation calculation

Let's say we have functional of the form: $$\mathcal{L} = \int_a^bF(y, y') dx$$ Let $\mathbf{x}(t) = (y(x,t), y'(x, t))^T$, then: $$\mathcal{L}(t + h) = \mathcal{L}(t) + h\frac{d}{dt}\mathcal{L}(t) +...
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2answers
104 views

Courant (1943) and History of Finite Element Method

I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal ...
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18 views

Definition of the functional inverse

In the context of the calculus of variations, I have seen the following used to define $V^{-1}$ as the "functional inverse" of a functional $V$: $$ \delta(x-y) = \int V(x,t|f)V^{-1}(t,y|f) dt \tag{1} ,...
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1answer
37 views

Is there a functional (i.e. infinite-dimensional) generalization of the second partial derivative test?

For a smooth function $f: \mathbb{R}^n \to \mathbb{R}$, we can (usually) test whether a critical point ${\bf x}_0$ (at which ${\bf \nabla} f({\bf x}_0) = {\bf 0}$) is a local maximum, minimum, or ...
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1answer
80 views

Euler-Lagrange equations for maximization problem

Given $\Omega:=[x_a,x_b]\times[y_a,y_b]\subset\mathbb{R}^2$, consider the problem $$ \max_{f\in \mathcal{C}^2(\Omega, \mathbb{R})} \iint_\Omega \left[f_y(x,y)\int_{x_a}^{x}f(z,y)\,dz\right]\, dx\, dy $...
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42 views

Extension of Du-Bois-Raymond lemma to Vector Fields on a Riemannian Manifold

I am trying to show the following extension of the Du Bois Raymond lemma: Let $M$ be a smooth Riemannian Manifold and $\omega: [0,1] \rightarrow M$ be a $W^{1,2}$ curve on M. Consider a tangential ...
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2answers
40 views

Finding extremals of a functional with integral function as integrand

Assume you want to find the extremals for the functional $$ y \rightarrow \int_a^by(x)\left[\int_a^xy(\xi)\, d\xi\right]\, dx $$ where $[a,b]\subset \mathbb{R}$ and $y\in \mathcal{C}^1\left([a,b],\...
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40 views

Is the functional $w \mapsto \int_0^1 | \ \| w(t) \| - 1 | \ dt$ $C^1$ or even smooth?

Let $H:= H(I;\mathbb{R}^3)$ be the space of $L^2$ + absolutely continuous functions with $L^2$ derivative. For $w \in H$ consider the functional $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ ...
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1answer
43 views

Show that the variational formulation has at most one solution

We have the problem: $$ -u''(x) + u(x) = f(x) ,\quad \quad x \in [0,L] $$ $$u(0) = 0 $$ $$u'(L) + u(L) = 4 $$ I then put it into variational form (hopefully correctly done) with introduction of $v \...
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57 views

Find the extremals of the functional $\int\sqrt{x^2+y^2}\sqrt{1+(y'(x))^2}dx$?

I want to use the polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$. After transformation, I get $$\int r\sqrt{r^2+(r')^2}d\theta.$$ Then, I derived the Euler-Lagrange equation $$2r^2-rr''+3(r')^2=...
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19 views

Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time. Context I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\...
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2answers
133 views

Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
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1answer
79 views

Why is compactness important to find solution of certain problems?

I've been recently re-thinking about my knowledge of compacts sets and I've realized that other than the definition and maybe understanding with some struggle some proofs I don't actually see why they'...
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0answers
35 views

Derivatives of the eikonal

Let $y$ be an extremal function of the functional $$J[y]=\int_{x_1}^{x_2} L(x, y(x), y'(x)) \mathrm{d}x$$ So it satisfies the Euler-Lagrange equation: $$\frac{\partial L}{\partial y}-\frac{\mathrm{d}}{...
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2answers
65 views

Is there a geometrical explanation for a method using critical points in a multivariable context

I would like to describe you a technique that I found in a physics paper, for which I cannot give a mathematical interpretation. Let $f(q,g,e,r)=\frac{r}{2}(g+q) + q^{\frac{p}{2}} e - \frac{T}{2} \...
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1answer
68 views

Existence of a minimizer by means of direct methodes

The following is given: Let $\Omega\subset \mathbb{R}^n$ be a bounded, connected open set with Lipschitz boundary. Let $f\in C^0(\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^n)$, $f=f(x,u,\xi)$,...
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1answer
138 views

proving an interpolation inequality in $L^p$ norms

Let $1\le p \le \infty$. Prove that for all $\epsilon >0$ there exists a constant $C>0$ such that $$\|u'\|_{L^p(\mathbb{R})}\le \epsilon \|u''\|_{L^p(\mathbb{R})}+C\|u\|_{L^p(\mathbb{R})} \;\; \...
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0answers
25 views

How to determine if an equation is an Euler-Lagrange equation of some functional?

Given an equation (e.g., a PDE), is there any way to determine if it is an Euler-Lagrange equation of some functional? If yes, is there in general any method to find out the functional? Or must there ...
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1answer
28 views

How are the steps to the solution for Arc - Length obtained?

Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? ...